X¯ Chart with Estimated Parameters: The Conditional ARL Distribution and New Insights

Performance measures of control charts with estimated parameters are random variables and vary significantly across reference samples. In this context, a recent idea has been to study the distribution of the realized (or conditional) in‐control average run length (CARL0) [or, equivalently, the conditional false‐alarm rate (CFAR)] for a set of estimates from a given reference sample and apply the exceedance probability criterion (EPC) to design control charts that ensure a desirable in‐control performance. Under the EPC, the probability that the CARL0 (or the CFAR) is at least (or at most) equal to a specified value is guaranteed with a high probability, which helps prevent low in‐control ARL's (or high false‐alarm rates) from occurring. In order to apply the EPC, the c.d.f. of the CARL0 (or the CFAR) is necessary. For the two‐sided Shewhart Xbar control chart, under normality, we derive the exact c.d.f. of the CARL0 and the CFAR, currently not available in the literature. Using these key results, we calculate the minimum number of Phase I samples required to guarantee a desired in‐control performance in terms of the EPC. Since the required amount of data can be prohibitively large, we also provide exact formulas for adjustments to the control limits for a given amount of Phase I data; some tables are provided. Our adjustment formulas give more accurate results compared to some available methods. The impact of these adjustments on the out‐of‐control performance of the chart is examined in detail. A summary and some recommendations are provided.